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Introduction to quantum computation YIA-CHUNG CHANG ( 張亞中 ) Research Center for Applied Sciences (RCAS) Academia Sinica, Taiwan (中研院 應用科學中心) Collaboration: University of Illinois Angbo Fang, Gefei Qian (Phys) theoretical modeling John Tucker (ECE) design of test structures Milton Feng (ECE) semiconductor processing Utah State University T.-C. Shen (Phys) STM donor patterning University of Utah Rui Du (Phys) low-T measurements Quantum Algorithm vs. Classical Algorithm Input: a single number a coherent superposition of many numbers Register: bit: 0, 1 qubit: 0 =, 1= or (a 0+b1) Processing: sequential massive parallel Time for solving QM: exponential linear e.g., One qubit operation, H: 0 (0 + 1)/√2 (Hadamard Transform) Do this on two qubits 0 0 (0 + 1)(0 + 1)/ 2 =(00 + 01+10 + 11)/ 2 The input now has 4 different binary numbers. Similarly, perform the H-transform on N qubits can generate 2N different binary numbers Applications: cryptography, data-base searching, teleportation,…etc. Classical v. s. Quantum Computation Information Unit Single-Bit NOT Gate Classical Bit: 0 or 1 NOT: 0 1 1 0 Quantum Qubit: 0+ 1 22 Unitary Operation 0+ 1 0+ 1 a, b a, ba A, B A, BA 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 Two-bit XOR Gate 00 00 01 01 10 11 11 10 Measurement Result: 0 or 1 0 with ||2 probability 100% certainty! 1 with ||2 probability U CNOT Any unitary operation on n qubits may be implemented exactly by composing single qubit and CNOT gates. The DiVincenzo Criteria To build a workable large-scale quantum computer, we need 1. 2. 3. 4. 5. 6. 7. A scalable physical system with well-characterized qubits. The ability to initialize the state of the qubits to a simple fiducial state, such as 000 . Long relevant decoherence times, much longer than the gate operation time. A “universal” set of quantum gates. A qubit-specific measurement capability. The ability to interconvert stationary and flying qubits. The ability to faithfully transmit flying qubits between specified locations. Principle of the SET transistor Like a MOSFET, the single-electron tunnelling (SET) transistor consists of a gate electrode that electrostatically influences electrons travelling between the source and drain electrodes. However, the electrons in the SET transistor need to cross two tunnel junctions that form an isolated conducting electrode called the island. Electrons passing through the island charge and discharge it, and the relative energies of systems containing 0 or 1 extra electrons depends on the gate voltage. An electron in a box (a) When a capacitor is charged through a resistor, the charge on the capacitor is proportional to the applied voltage and shows no sign of quantization. (b) When a tunnel junction replaces the resistor, a conducting island is formed between the junction and the capacitor plate. In this case the average charge on the island increases in steps as the voltage is increased (c). The steps are sharper for more resistive barriers and at lower temperatures. Advantages of Quantum Algorithms Complexity Factoring an n-bit number Searching M solutions out of N possibilities Classical 1/ 3 exp[ cn (ln n) N M Quantum 2/3 2 n (ln n)(ln ln n) (Shor) ] N M (Grove) Basic Procedure of Quantum Computation 1. Prepare an appropriate initial state in N-qubit space 2. Implement the desired quantum algorithm via a series of elementary gate operations 3. Measure the final state in an appropriate basis and extract the solution from measurement result by some simple classical computation